4 ANALYTICAL MECHANICS is'moved about without changing its direction and magnitude. 7. Addition of Two Vectors.—Let the vectors a and b, 1< ig. 2, represent two displacements, then their sum is another vector, c, which is equivalent to the given vectors. In order to find c let us apply to a particle the operat ions indicated by a and b. Each vector displaces the particle along its direction through a distance equal to its length. There- fore applying a to the particle at P, Fig, 3, the particle is brought to the point Q. Then applying the. operation indicated by b the particle is brought to the point It Therefore the result of the two operations is a displacement from P to R. But this is equivalent to a single operation represented by the vector c, which has P for its origin and H for its terminus. Therefore c is called the sum, or the result ant, of a and b. This fact is denoted by the following vedor equation, a + b-c. (!) 8. Order of Addition. — The order of addition clow not affect the result. If in Fig. 3 the order of the operations indicated by a and b is reversed the particle moves from P to (/ atid then to R. Thus the path of the particle is changed hut not the resultant displacement. 9. Simultaneous Operation of Two Vectors* The operations indicated by a and b may be performed simultaneously